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Global solutions of nonlinear Schrödinger equations. (English) Zbl 0933.35178
Colloquium Publications. American Mathematical Society (AMS). 46. Providence, RI: American Mathematical Society (AMS). viii, 182 p. (1999).
The theory of nonlinear dispersive equations has received a tremendous amount of attention in the last few years. This interest has resulted in a number of important developments. Among the equations that have been in the forefront of such studies one finds the nonlinear Schrödinger (NLS) equation: \[ i u_t + \Delta u \pm u |u|^{p-2} = 0 . \] The qualitative behavior of the NLS solutions is heavily dependent upon the sign that goes with the nonlinear term. In the plus sign case, smooth solutions of the NLS equation with domain in \({\mathbb R}^d\) may blow up in finite time for \(p \geq 2 + (4/d)\). In the minus case, which is called the defocusing case, one expects to extend the solutions to global ones while still preserving their \(H^{s}\) class.
The monograph under review, which was written by one of the key researchers in this field, is concerned with several aspects of the Cauchy problem for the NLS in different Sobolev spaces. In particular, it discusses the defocusing critical NLS with radial data. It presents new techniques and results on global existence of large data solutions below the energy norm. It discusses current research connected to Strichartz type inequalities and its relevance to nonlinear partial differential equations. It is also concerned with nonlinear wave equations, invariant Gibbs measures associated to the flow of the NLS equation, and Kolmogorov-Arnold-Moser (KAM) theory.
The work starts with an introductory chapter, where some of the main topics that will be addressed in the monograph are described. Then, it presents an overview of related results on NLS type equations. Several results on wellposedness of the Cauchy problem for different spaces, and scattering follow. In Chapter III the author studies the so-called \(H^{1}\) critical defocusing of the three-dimensional NLS equation. Chapter IV concerns the problem of wellposedness below the energy norm both for nonlinear wave (NLW) equations and for the defocusing NLS. Chapter V considers the NLS equation on bounded domains, mostly with periodic boundary conditions. This chapter starts with sections on the Cauchy problem and periodic Strichartz inequalities. Then, it proceeds with the topic of the construction of invariant Gibbs measures for the NLS flows both in one dimension and in the multidimensional case. After surveying a number of results on the periodic case, the author considers the construction of invariant Gibbs measures for unbounded domains. The last section of this chapter deals with quasi-periodic solutions of the NLS.
The monograph has two appendices. The first one dealing with the growth of the Sobolev norm for the linear Schrödinger equation with smooth time dependent potentials. The second one on the so-called Zakharov systems, which are obtained coupling NLS and NLW equations.
Despite the recent progresses, the field is far from being exhausted. In fact, as the author clearly demonstrates throughout the monograph, there are a number of interesting open problems. Some small typos could be detected in the text, but this is minor when one considers the importance of publishing a timely monograph on this rapidly growing field.
Summing up, the text under review is an excellent exposition of the recent results on global solutions of nonlinear Schrödinger equations and related topics such as Strichartz inequalities, invariant Gibbs measures, and recent developments in KAM theory for partial differential equations. The stress on current research topics and open problems will most certainly make this monograph a classic.

35Q55 NLS equations (nonlinear Schrödinger equations)
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
37K55 Perturbations, KAM for infinite-dimensional Hamiltonian and Lagrangian systems